which has a simple interpretation9: P and B share the expected surplus given by the sum of the net expected gain of the briber (Cx(r)is the amount of avoided compliance costs, ranging from 0 to CX) and the expected penalty (reduction in utility) of the bribee. Since ∂b / ∂x ≥ 0 for f P ≤ (1 − Θ) / Θ, bfP 0 and bfB 0, by implicit differention we can see that dfP/dx 0 and d2fP/dx2 0 (concavity of the privates’ penalty), dfB/dx 0 and dfP/dfB 0.

Equation (5) tell us that the condition for b 0, i.e. the condition for corruption in the society, requires Θ   Cx(r ) + 1 − Θ ( f B − Cx(r ) f P )  0 for 0 ≤ x(r ) ≤ X   Call C1 the value of C that satisfies the above equation as an equality given the parameters. Then F[C] evaluated at any C C1 gives the probability of the realization of the compliance costs that makes corruption equal zero10. We use it11 in the following modified version of Problem 1. The government assumes that when costs are in the interval [C, C1 ] there is no bribe, given Θ, fP and fB. Therefore, the problem is to issue the efficient amount of regulation and determine the optimal fine functions that induce the required compliance when unobservable expected costs are assumed to be greater than C1 and bribes are therefore possible. Using the above utility function of the privates the government calculates, for C C1, by the Envelope Theorem, * dU P = − X + x(r ) − Θx(r ) f P dC b has a lower bound of zero for x = 0 (Θ=0), and an upper bound of CX + (θ /1 − θ )[ f B − CXf P ) for x = X.

Making fB dependent on w (i.e. fB = γw, with 0 γ 1) penalizes seniority with respect to the case of a lump-sum fine, provided that salary increases with years in office. The effect is to increase the threshold value of C1 discussed in the text for high rank officials thereby making corruption less likely for these kind of officers: a result that is generally claimed in the literature for high wage officers.

Reference to the stake of corruption to define a probability of corruption in the model is used also in Auriol et al (2011, 218) to compare the probability of corruption in alternative ownership regimes after SOEs privatization.

and solves, under the assumption that x can be observed but its corresponding costs are not, the welfare maximization problem under the above incentive compatibility constraint for the privates12

– &nbsp– &nbsp–

The last condition gives, upon integration between C1 and C, µ = −[1 − F (C1 )] for any C1 C and therefore, by substitution, the optimal fine per unit of incompliance (index 1 dropped) is

– &nbsp– &nbsp–

where dr/dx = (dx/dr)-1 Then, the optimal fine per unit of incompliance has two components: a fix part plus or minus the gain/loss generated by the sign of the reciprocal of dx/dr (marginal incompliance). When dx/dr = 0 privates do not modify their decision on incompliance whatever the regulation policy and the fine is a constant.

We call this situation neutral reaction to regulation. When dx/dr 0 an increase in r induces more incompliance on the part of privates (adverse reaction to regulation) and therefore the fine has to include the gain potentially generated by an efficient choice of r (i.e. V’ – w’) that is forgone because of the privates’ reaction. Privates should be punished by including in the fine the foregone welfare increase. When dx/dr 0 (accommodating reaction to regulation) an increase in r implies that privates progressively reduce incompliance as if compliance were complementary to regulation and therefore their perspective fine can be reduced: they should benefit from their virtuous behaviour. For dx/dr 0 the penalty becomes a subsidy when

Notice the role played by the hazard rate of the distribution in the above result: it makes the realization of the positive welfare gain ( V’ – w’) more or less likely in terms of the conditional density of non complying behaviour at costs C C1 given that privates have complied (see eq. 5) at costs at or below C1.

Since (5) is valid for any value of C C1 we can substitute (5) into (6) to obtain, for any value of the unknown costs, the penalty for the officer for any C C1 consistent with b 0:

– &nbsp– &nbsp–

As one can verity, substituting (6) and (7) into (5) leads, other things being equal, to a zero bribe payment for any value of C C1. Moreover, notice that the penalty for B is always positive if dx/dr 0 i.e. when privates increase incompliance and negative when

– &nbsp– &nbsp–

for dx/dr 0 i.e. when privates reduce incompliance. Notice once again the role of the hazard rate of F(C) which is analogous with respect to the case of fP.

When the government’s policy induces compliance, fP reduces in (6). Privates may appropriate part the results of their reaction to regulation. Then, although no specific conclusion for the value of penalty of the officers can be drawn, still we can say that their penalty should be smaller than fP and should have the same property of fP with respect to the sign of dr/dx.

3.3 Corruption and the allocative cost of bribes

Rose–Ackerman et al. (2012, 6) suggested that bribes, like a distorsive tax, are not simple inter-individual lump-sum transfers. On the contrary, they affect allocative efficiency as if, in our interpretation, something similar to the marginal costs of public funds were attached to a bribe as it is to any indirect tax. As such, bribes produce a reduction of welfare that should be related to dx/dr and as such charged to corrupted parties. We show that this is indeed the case.

Assume the shadow cost of bribery is defined in terms of bureaucracy salary so that that D(b) = D + λw(r)b with dD/dr = λw’b 0. Substituting for D in the H function above and maximizing it under the same incentive compatibility constraint gives the gain from regulation reduced by λ (the shadow cost of corruption) times the amount of the total bribe. Solving the modified problem we obtain

– &nbsp– &nbsp–

which means that the distortionary cost of corruption must have an increasing effect on the fine (the numerator of the term in braces is higher) that must affect fP through dr/dx and the hazard rate. Once again, the reaction to regulation can be accommodating or adverse in terms of enhancing compliance and, under the hypothesis that bribes induce a distortion in the allocation of resources (λ 0), the effect of λb on fP depends upon the effect of regulation on incompliance and the hazard rate, i.e. upon how likelihood is that C C1 (greater potential gain from corruption). Hence, the resulting higher value of λb which reduces the net marginal gain of regulation must be paid by privates according to the value of dr/dx.

3.4 Discussion and relations with the literature

Reference to compliance costs and government regulation has permitted to present a somewhat general framework for compliance and corruption analysis. The existing literature, on the contrary, discusses separate and specific cases of corruption decisions (procurement, police abuse, taxes, ecc.). Still, the results obtained in the previous sections can be related to the existing literature. First, notice that in our setting above, investment in r is an investment in “enforcing by monitoring” i.e. an activity that, likewise in Mookherjee et al. (1992), the regulator implements by committing resources before receiving information about the offence, if any. This should not be confused with the activity of enforcement by investigation. If in (4) w is interpreted as the ongoing salary prevailing in the economy, the payment structure represents a variant of those efficiency wage payments discussed by Ades et al. (1997, 504), among others. Assume, as an example, that D is the loss from fees unpaid to the government for some private activity (social contributions, in the reported example of Ades et al (1997)), the last part of (4) incorporates in the officers’ salary part of the fees collected by the activity of the bureau whereas the second component of the salary is the marginal gain generated by their activity. Under symmetric information on compliance costs and results of the bureaucratic monitoring activity, which implies that bribes are not possible, officers can be made residual claimants of the results of their work. Concavity of the fine function parallels Rose–Ackerman (1975, 193) result that concavity is “consistent with a sanctioning strategy under which the penalty upon conviction is solely a function of the size of the bribe paid and the probability of conviction is a concave function of the firms’ revenues”. In our model regulation/monitoring implies to enforce efficient compliance – or discovery of ex-post non compliance – requires, absent bribes, concavity in r and D. As for our result when bribing is possible, we stress that Mookherjeen et al (1995) obtain a result similar to that shown by our equation (6) and (7), i.e. that it should be punished more the bribe giver than the receiver, and Lambsdorff (2007) suggests this measure should be adopted when the receiver fails to reciprocate after taking the bribe.

As for the fines of public officers, we have followed the literature and supposed that the officers are not fired and that the penalty per unit of incompliance is flat. Analogously we have supposed that the private, upon discovery, is not entirely deprived from the results of his non complying action. We have assumed that he/she is forced to fully comply and fined with a flat rate penalty. Even with this simplifying assumptions, however, the model permits to conclude that a permanent reduction of the salary or a fine increasing with the wage would make corruption less likely, at least for high rank officials (see footnote 9). This accords with results on efficiency wages obtained by previous literature. Indeed, also in our case the smaller the advantage from curbing corruption the lower the incentive part of the wage to be paid to officers. Still, we have shown that the probability of being discovered is not the exclusive factor that affects corruption decisions, even when moral costs are not taken into account.

We found that simplification of privates’ compliance activities affects the critical value of C and reduces both incompliance and bribes, not to mention the allocative cost of corruption. A similar result was obtained empirically only for utilities operating in developing countries by Seim et al. (2009) and more recently by Johnson et al. (2013) for regulatory regimes in some US states but no previous theoretical work has analysed a model build on this idea. We provide a general framework in which compliance costs could be associated to the income level of bribers and bribees (through officers’ wage and, indirectly, through the monetary value of the regulated action of the privates) in order to incorporate agents’ income level into the model. We have shown that fines should be independent of such variables and this makes anti-corruption measures seemingly neutral in distributional terms.

The allocative cost of corruption (opportunity costs of the bribes) is reflected in the fines’ formulas and this characterizes our results with respect to other models in which efficiency requires some equilibrium level of corruption.

Our main result is that compliance might not always be the most efficient choice from a social point of view and therefore that there can exist an equilibrium level of corruption.

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